Explicit lower bounds for linear forms

Abstract:

Let $\mathbb {I}$ be the field of rational numbers or an imaginary quadratic field and $\mathbb {Z}_\mathbb {I}$ its ring of integers. We study some general lemmas that produce lower bounds \[ \lvert B_0+B_1\theta _1+\cdots +B_r\theta _r\rvert \ge \frac {1}{\max \{\lvert B_1 \rvert ,\ldots ,\lvert B_r \rvert \}^\mu } \] for all $B_0,\ldots ,B_r \in \mathbb {Z}_{\mathbb {I}}$, $\max \{\lvert B_1 \rvert ,\ldots ,\lvert B_r \rvert \} \ge H_0$, given suitable simultaneous approximating sequences of the numbers $\theta _1,\ldots ,\theta _r$. We manage to replace the lower bound with $1/{\max \{\lvert B_1 \rvert ^{\mu _1},\ldots ,\lvert B_r \rvert ^{\mu _r}\}}$ for all $B_0,\ldots ,B_r \in \mathbb {Z}_{\mathbb {I}}$, $\max \{\lvert B_1 \rvert ^{\mu _1},\ldots ,\lvert B_r \rvert ^{\mu _r}\} \ge H_0$, where the exponents $\mu _1,\ldots ,\mu _r$ are different when the given type II approximating sequences approximate some of the numbers $\theta _1,\ldots ,\theta _r$ better than the others. As an application we research certain linear forms in logarithms. Our results are completely explicit.